Lecture 3: Edge detection

CS4670/5670: Computer VisionKavita Bala

From Sandlot Science

Announcements

• Find partners on piazza

• PA 1 will be out on Monday

• Quiz on Monday or Wednesday, beginning of class

Why edges?

• Humans are sensitive to edges• Convert a 2D image into a set of curves

– Extracts salient features of the scene, more compact

Origin of Edges

• Edges are caused by a variety of factors

depth discontinuity

surface color discontinuity

illumination discontinuity

surface normal discontinuity

Images as functions…

• Edges look like steep cliffs

Characterizing edges• An edge is a place of rapid change in the

image intensity function

imageintensity function

(along horizontal scanline) first derivative

edges correspond toextrema of derivativeSource: L. Lazebnik

• How can we differentiate a digital image F[x,y]?– Option 1: reconstruct a continuous image, f, then

compute the derivative– Option 2: take discrete derivative (finite difference)

1 -1

How would you implement this as a linear filter?

Image derivatives

-1

1: :

Source: S. Seitz

The gradient points in the direction of most rapid increase in intensity

Image gradient• The gradient of an image:

The edge strength is given by the gradient magnitude:

The gradient direction is given by:

• how does this relate to the direction of the edge?Source: Steve Seitz

Image gradient

Source: L. Lazebnik

Effects of noise

Where is the edge?Source: S. Seitz

Noisy input image

Solution: smooth first

f

h

f * h

Source: S. SeitzTo find edges, look for peaks in

• Differentiation is a convolution• Convolution is associative:• This saves us one operation:

Associative property of convolution

f

Source: S. Seitz

2D edge detection filters

Gaussianderivative of Gaussian (x)

Derivative of Gaussian filter

x-direction y-direction

The Sobel operator• Common approximation of derivative of Gaussian

– A mask (not a convolution kernel)

-1 0 1

-2 0 2

-1 0 1

1 2 1

0 0 0

-1 -2 -1

• The standard defn. of the Sobel operator omits the 1/8 term– doesn’t make a difference for edge detection– the 1/8 term is needed to get the right gradient magnitude

Sobel operator: example

Source: Wikipedia

Questions?

Image Resampling

CS4670: Computer Vision

Image Scaling

This image is too big to fit on the screen. How can we generate a half-sized version?

Source: S. Seitz

Image sub-sampling

Throw away every other row and column to create a 1/2 size image

- called image sub-sampling

1/4

1/16

Source: S. Seitz

Image sub-sampling

1/4 (2x zoom) 1/16 (4x zoom)

Why does this look so crufty?

1/2

Source: S. Seitz

Image sub-sampling

Source: F. Durand

Even worse for synthetic images

Source: L. Zhang

What is aliasing?

• What if we “missed” things between the samples?• Simple example: undersampling a sine wave

– unsurprising result: information is lost– surprising result: indistinguishable from lower frequency– also was always indistinguishable from higher frequencies– aliasing: signals “traveling in disguise” as other frequencies

28

Wagon-wheel effect

(See http://www.michaelbach.de/ot/mot_wagonWheel/index.html)Source: L. Zhang

Aliasing

• Occurs when your sampling rate is not high enough to capture the amount of detail in your image

• Can give you the wrong signal/image—an alias• To do sampling right, need to understand the structure of

your signal/image• Enter Monsieur Fourier…• To avoid aliasing:

– sampling rate ≥ 2 * max frequency in the image• said another way: ≥ two samples per cycle

– This minimum sampling rate is called the Nyquist rate

Source: L. Zhang

Nyquist limit – 2D example

Good sampling

Bad sampling

Aliasing

• When downsampling by a factor of two– Original image has frequencies that are too high

• How can we fix this?

Gaussian pre-filtering

G 1/4

G 1/8

Gaussian 1/2

• Solution: filter the image, then subsample

Source: S. Seitz

Subsampling with Gaussian pre-filtering

G 1/4 G 1/8Gaussian 1/2

• Solution: filter the image, then subsample

Source: S. Seitz

Compare with...

1/4 (2x zoom) 1/8 (4x zoom)1/2

Source: S. Seitz

Gaussian pre-filtering

• Solution: filter the image, then subsample

blur

F0 H*

subsample blur subsample …F1

F1 H*

F2F0

blur

F0 H*

subsample blur subsample …F1

F1 H*

F2F0

{Gaussian pyramid

Gaussian pyramids [Burt and Adelson, 1983]

• In computer graphics, a mip map [Williams, 1983]• A precursor to wavelet transform

Gaussian Pyramids have all sorts of applications in computer visionSource: S. Seitz

Gaussian pyramids [Burt and Adelson, 1983]

• How much space does a Gaussian pyramid take compared to the original image?

Source: S. Seitz

Gaussian Pyramid

Questions?